US6093211A  Polymer property distribution functions methodology and simulators  Google Patents
Polymer property distribution functions methodology and simulators Download PDFInfo
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 US6093211A US6093211A US09057795 US5779598A US6093211A US 6093211 A US6093211 A US 6093211A US 09057795 US09057795 US 09057795 US 5779598 A US5779598 A US 5779598A US 6093211 A US6093211 A US 6093211A
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Abstract
Description
Numerous articles (particularly those of synthetic material such as plastics and rubber) are produced by a respective chemical process, namely polymerization. Each such chemical process/polymerization involves a complex chemical reaction brought about in a reactor or network of reactors and ancillary equipment (e.g., distillation towers, dryers, extruders, etc.) Typically, the chemical process and polymer manufacturing systems (i.e., reactors and ancillary equipment) are expressed in terms of formulas or equations for purposes of modeling or simulating the chemical process/polymer manufacturing system.
The demands on simulation models for polymerization reactor processes and polymer manufacturing systems in general is growing due to advanced technology for polymerization catalysts such as metallocene catalysts. There is a rapidly growing need for simulation models to calculate full distributions of composition, molecular weight and long chain branching. This presents serious problems related to information transfer efficiency for both steadystate and dynamic simulations of polymer reactor systems which are part of a larger polymer manufacturing system.
Commonly the population balance equations of the subject polymer manufacturing system are solved by mathematically integrating either solving for species or solving for moments of distribution. The solution of the very large sets of algebraic and ordinary differential equations which comprise the population balance equations appears impossible even with the largest and fastest computers available today. Another problem which presents itself is the need to transfer concentration data of the very large number of polymer species (perhaps as many as 10^{5}) through the network of polymer reactors and other process units in the polymer manufacturing system.
An efficient simulation strategy which gives high information transfer efficiency employs the method of using moments of the distribution (called the "method of moments") to calculate average polymer properties. (See U.S. Pat. No. 5,687,090, issued Nov. 11, 1997 to assignee of the present invention). Although the strategy satisfies many demands of polymer process simulators, it has one weakness which cannot be overcome without a complete change in polymer process/reactor modeling strategy. When polymer distributions in composition, molecular weight and long chain branching are complex (having a shoulder or bimodal, for example), these distributions cannot be accurately constructed (or reconstructed) given the moments of these distributions. In fact, the process of constructing complex distributions given moments (as many as one wishes) leads to multiple solutions and possible large errors.
In contrast to the method of moments, instantaneous property methods have been studied in isolated incidences. W. H. Stockmayer in J.Chem.Phys. 13:199 (1945) derives and discloses a useful analytical expression for bivariate distribution of properties (e.g., polymer chain chemical composition and molecular weight) for dead polymer chains produced instantaneously. However, he does not address or contemplate the use of such instantaneous property measures to track distribution of polymer properties (such as composition, molecular weight and long chain branching) through polymer reactor networks for modeling/simulation of reactor networks or polymer manufacturing systems.
A. H. AbdelAlim and A. E. Hamielec in the J. Applied Polym. Sci., 16, 783 (1972) show the use of instantaneous property methods as applied to the batch polymerization of vinyl chloride. This paper was referred to in a text by J. A. Biesenberger and D. H. Sebastian, entitled "Principles of Polymerization Engineering," WileyInterscience, New York, (1983) in which instantaneous property methods were discussed.
A. E. Hamielec's research associates and Ph.D. students dominated the use of instantaneous property methods and these applications have all been for single batch polymerization reactors. Some of these publications include:
S. T. Balke and A. E. Hamielec, "Free Radical Polymerization of Methyl Methacrylate to High Conversions," J. Applied Polym. Sci., 17:905 (1973).
L. H. GarciaRubio and A. E. Hamielec, "Bulk Polymerization of AcrylonitrileModel Development," J. Applied Polym. Sci., 23:1413 (1979).
C. J. Kim and A. E. Hamielec, "Polymerization of Acrylamide with DiffusionControlled Termination," Polymer, 25:845 (1984).
O. Chiantore and A. E. Hamielec, "Thermal Polymerization of pMethyl Styrene at High Conversions and Temperatures," Polymer, 26:608 (1985).
D. Bhattacharya and A. E. Hamielec, "Bulk Thermal Copolymerization of Styrene/pMethyl Styrene," Polymer, 27:611 (1986).
K. M. Jones et al., "An Investigation of the Kinetics of Copolymerization of Methyl Methacrylate/pMethyl Styrene," Polymer, 27: 602 (1986).
T. M. Yaraskavitch et al., "An Investigation of the Kinetics of Copolymerization of pMethyl StyreneAcrylonitrile," Polymer, 28:489 (1987).
S. Zhu and A. E. Hamielec, "Chain Length Dependent Termination for FreeRadical Polymerization," Macromolecules, 22:3093 (1989). This paper is notable for its generalization of the SchulzFlory and Stockmayer's distributions to account for chainlength dependent termination for freeradical polymerization.
S. Thomas et al., "FreeRadical PolymerizationAn Elegant Method of Solving the Population Balance Equations with Chain Transfer to Polymer" Polymer Reaction Engineering, 5:183 (1997). This reference is notable for the first attempt to use instantaneous property methods with long chain branching and tracking distributions of molecular weight and long chain branching in a batch reactor.
"Predici" a simulation of polyreaction kinetics product by CiT GmbH is marketed as providing the computation of complete molecular weight distribution of polymers in a process simulator/modeler. This product is an example of the method where population balance equations are solved as differential equations. See "The Simulation of Molecular Weight Distributions in Polyreaction Kinetics by Discrete Galerkin Methods," Macromol. Theory Simul., 5:393416 (1996).
Thus, to date, process modeling of polymer manufacturing systems remains computationally complex, inaccurate and inefficient or otherwise problematic.
The present invention overcomes the shortcomings of the prior art. In particular, the present invention does not attempt to solve the population balance equations as ordinary differential equations for polymerizations under transient conditions as is done in the prior art. The prior art solves ordinary differential equations for the population balance equations or for the moments of the distribution functions described by the population balance equations.
The present invention solves the differential form of the population balance equations as a set of algebraic equations to calculate the full distributions of composition, molecular weight, stereoregularity and long chain branching for polymer chains produced "instantaneously" or in a small time interval. This is called the "instantaneous property method," and the polymer chains produced instantaneously are referred to herein as a "pulse" of dead polymer. In addition, the present invention provides novel and efficient methods of tracking the mass and the distribution functions of polymer chains produced "instantaneously" in transient or steadystate polymer manufacturing systems which include reactor networks.
With the "instantaneous property method", the calculated distribution functions for polymer composition, molecular weight, stereoregularity and long chain branching are often analytical expressions with one or two kinetic parameters. The mass of the polymer produced "instantaneously" and its associated one or two kinetic parameters may be readily tracked in a polymer manufacturing system and when desired these kinetic parameters may be used to reconstruct the full distributions of composition, molecular weight, stereoregularity and long chain branching at any point of interest in the transient polymer manufacturing system including the polymer product stream leaving the manufacturing system. This gives no loss of information as with the method of moments and yields high information transfer efficiency.
As such, the present invention provides property distributions of composition, molecular weight, long chain branching, stereoregularity (and other polymer properties), for modeling reactor networks or polymer manufacturing systems, as heretofore unachieved in the prior art. By employing the instantaneous property measurements instead of the moment of distributions of the prior art, the present invention provides accurate and computationally efficient modeling of polymer processes and manufacturing systems for carrying out the same.
As used herein "polymer manufacturing systems" means a system including one or more polymer reactors and ancillary equipment (such as distillation columns/towers, dryers, extruders, etc.).
"Pulse of dead polymer" or a "pulse" refers to the polymer chains produced instantaneously. "Pulses" are generated at the active sites of polymerization catalysts. Some catalysts may possess a number of active sites with different polymerization characteristics. In other words, different types of polymer pulses (i.e., with different molecular properties) can be generated from different types of active sites. These different types of active sites are grouped as "site types".
"Polymer properties" includes chemical and physical properties. More specifically, polymer properties includes (a) molecular properties of polymers (e.g., distributions of composition, molecular weight, long chain branching and stereoregularity), (b) rheological or processing properties of polymers (e.g., zero shear melt viscosity, melt index, melt flow ratio, melt strength), and (c) mechanical properties (e.g., tensile modulus, elastic modulus, impact strength and environment stress cracking).
In the preferred embodiment, the present invention provides a method and apparatus for modeling a polymer manufacturing system. The polymer manufacturing system is formed of plural chemical reactors for carrying out a polymerization process, such as chain growth polymerization. The invention method comprises the steps of:
providing indications of reactor type and polymers involved in the polymerization process, each polymer having at least one molecular property (preferably with a distribution function such as composition, molecular weight, stereoregularity or long chain branching);
calculating or otherwise determining instantaneous molecular property measures of the properties of the polymers (including for example molecular weight, composition, stereoregularity and long chain branching distributions);
for at least one polymer property (e.g., one of composition, molecular weight, stereoregularity, and long chain branching frequency), calculating and tracking the property distributions and mass of the pulse from the calculated instantaneous kinetic parameters and subsequently calculating the full distribution of accumulated polymer of the property at any point in the polymer manufacturing system including the polymer product stream leaving the manufacturing system; and
using the calculated property distribution, representing the polymerization process such that the polymer manufacturing system is modeled.
In accordance with one aspect of the preferred embodiment, the end user specifies a reactor type of one or more batch, semibatch or continuous reactors. For continuous reactors, the preferred embodiment enables user specification of steady state or transient for modeling the same.
Further, the present invention provides tracking of polymer properties throughout a polymer manufacturing system for purposes of simulating the same or a polymer process. The invention tracking of polymer properties enables calculation of full property distributions in the model/simulation. The tracked polymer properties include mass, molecular weight, composition, stereoregularity and long chain branching.
The foregoing and other objects, features and advantages of the invention will be apparent from the following more particular description of preferred embodiments and the drawings in which like reference characters refer to the same parts throughout the different views. The drawings are not necessarily to scale, emphasis instead being placed upon illustrating the principles of the invention.
FIG. 1 is a schematic diagram of a reactor system which is modeled by the present invention.
FIG. 2 is a flow diagram of the software routine implementing a preferred embodiment of the present invention.
FIG. 3 is a block diagram of a computer system employing the software routine of FIG. 2.
By way of background and introduction to the specifics of the present invention, discussed next are the following:
Molecular Property Distribution Types
Distribution Functions
Distributions in Process Models
Average Properties and Moments
Method of Instantaneous Properties
Mechanism for Tracking Distributions
After discussing the foregoing, an example of the present invention method and apparatus for simulating/modeling polymer processes of reactor networks using the method of instantaneous properties (to calculate property distributions) is provided.
Molecular Property Distribution Types:
The common polymer molecular properties for which distributions are typically considered include
chain size (e.g., molecular weight or chain length)
copolymer composition
degree of branching
polymer particle size (for certain polymerization reaction types)
stereoregularity
In order to accurately characterize a polymer component, and maintain control of polymer product rheological and mechanical properties, physicists, chemists and engineers must concern themselves with most of these distributions.
From a modeling standpoint, many theoretical and empirical functions have been developed to represent distributions. Theoretical distribution functions are generally analytical solutions of the population balance equations. Some of these are reviewed here. These functions tend to fall into categories derived from their formulation, or from their graphical representation. For example, distributions which consider two dependent parameters simultaneously, e.g., chain size and copolymer composition, are termed "bivariate" distributions. Distributions which mimic the normal bellshaped graphical representation are called "unimodal" distributions. This is in contrast with distributions which reveal several peaks and are called "bimodal" or "multimodal" distributions.
Distribution Functions:
In the majority of cases, the distribution functions proposed in the literature are based on a statistical approach and use one of three types of mathematical functions: binomial, Poisson and Gaussian. The parameters in these distribution functions can easily be calculated from the polymer average properties (degree of polymerization, polydispersity index, etc.) The following are the common distribution functions which have been applied to the calculation of polymer molecular property distributions.
SchulzFlory Most ProbableSee
Flory, P. J., "Molecular Size Distribution in Linear Condensation Polymers," J. Am. Chem. Soc., 58:1877 (1936).
Flory, P. J., Principles of Polymer Chemistry, Cornell University Press, Ithaca, N.Y. (1953).
Schulz, G. V., "Uber die Bezeihung zwischen Reaktionsgeschwindigkeit und Zusammensetzung des Reaktionsproduktes bei Macropolymerisationsvorgangen," Z.Physik.Chem., B30:379 (1935).
Schulz, G. V., "Uber die Kinetik der kettenpolymerisationen. V Der Einfluss verschiedener Reaktionsarten auf die Polymolekularitat, Z. Physik. Chem, B43:25 (1939).
Schulz (Schulz, 1935/1939)
WeibullTung Generalized ExponentialSee
Weibull, W., "A Statistical Distribution Function of Wide Applicability, J. Appl. Mech., 18:293 (1951).
Tung, L. H., "Fractionation of Polyethylene, J. Polymer Sci., 20:495 (1956).
NormalSee
Biesenberger, J. A. and Sebastian, D. H., "Principles of Polymerization Engineering," WileyInterscience, New York (1983).
Wesslau Logarithmic NormalSee
Wesslau, H., "Die Molekulargewichtsverteilung einiger Niederdruckpolyathelene," Makromol. Chem, 20:111 (1956).
Lansing Logarithmic NormalSee
Lansing, W. D. and Kramer, E. O., "Molecular Weight Analysis of Mixtures by Sedimentation Equilibrium in the Svedberg Ultracentrifuge," J. Am. Chem. Soc., 57:1396 (1935).
PoissonSee
Biesenberger (1983).
ZimmSee
Zimm, B. H., "Apparatus and Methods for Measurement and Interpretation of the Angular Variation of Light Scattering; Preliminary Results on Polystyrene Solutions," J. Chem. Phys., 16:1099 (1948).
Stockmayer BivariateSee
Stockmayer, W. H., J. Chem. Phys., 13:199 (1945).
In addition to these distribution functions, a method using the moments of distributions is available (Tompa, H., "The Calculation of MoleWeight Distributions from Kinetic Schemes," Chap. 7 in Comprehensive Chemical Kinetics," Vol. 14A, C. H. Bamford and C. F. H. Tipper, Eds, American Elseview, New York (1976)). Two of these functions have greater importance for the present invention. These are now discussed in more detail.
SchulzFlory Most Probable Distribution
Schulz and Flory developed a oneparameter equation to represent the distribution of polymers which fall into one of the following categories.
addition polymers formed by a constant rate of initiation, with invariant monomer concentration, with termination by disproportionation only, and with no chain transfer to monomer
linear condensation polymers obeying the assumption of equal reactivities of chain ends or linear condensation polymers formed by random interchange of units
low molecular weight polymer formed from a high molecular weight polymer by random scission
The Schulz/Flory distribution is also known as the MostProbable distribution since it is dictated by the probability of random events, such as the location of a scission reaction on a longchain molecule. It should be pointed out that the limitation for the use of the SchulzFlory distribution, i.e., constant rate of initiation with invariant monomer concentration is no longer in effect when this distribution function is used with the "instantaneous property method" of the present invention. The number or molefraction distribution and the weight fraction distribution are given by
F(r)=p^{r1} (1p) for number distribution, and
W(r)=rp^{r1} (1p)^{2} for weight distribution
where p is the extent of reaction with r being the size of the molecule or number of segments; for addition polymerizations p is the probability that a growing live polymer molecule will propagate. For stepgrowth reactions, p is the fractional conversion of monomer end groups.
From these distributions, the number average (n), weight average (w) and z average (z) degree of polymerization (DP), and the polydisperisty index (PDI) are ##EQU1##
PDI=1+p
To generate the distribution, p is calculated from degree of polymerization (DP) as ##EQU2## Note that as p goes to 1, PDI goes to 2. Stockmayer Bivariate Distribution
There are cases where two polymer property distributions must be considered simultaneously. Functions which consider two properties simultaneously are called bivariate. Stockmayer solved the population balance equations and developed such a distribution function to consider both chain size and composition distribution for example (Stockmayer, 1945). This model may be extended to other combinations of polymer properties such as chain size and long chain branching distribution for the case of copolymers.
Distributions in Process Models:
In order to track property distributions within the framework of a process simulation package (computer program), one approach would be to implement one of the distribution functions discussed earlier, and store the calculated values for the distributions throughout the simulation. This approach would require large amounts of data storage and would take unreasonably long CPU times. For this reason, only average polymer properties were tracked in prior art process modeling software.
There is however, a great demand to know the full molecular weight distribution, particularly for complex distributions that may have a shoulder, or are even bimodal. This information is needed for the purposes of optimization of rheological and mechanical properties of the final polymer product.
Within the present invention method and apparatus of process modeling, a dual approach for determining polymer properties is used. The method of moments continues to be the preferred approach for calculating average properties, while the method of instantaneous properties is used to calculate property distributions (e.g., molecular weight distribution, composition distribution and long chain branching distribution, and the like). The present invention method addresses the issue of data storage and computational complexity in tracking distributions. Under special circumstances, the most general form of the instantaneous distribution function reduces to Flory's most probable distribution. The instantaneous distribution functions are unimodal. However, the distribution functions for polymer accumulated in a multireactor system may be multimodal.
Average Properties and Moments
It is usually very convenient to examine polymer molecular properties in terms of averages instead of considering the complete distribution. These average properties must be determined from the actual distributions either through distribution moments or through instantaneous properties.
The average properties tracked for polymers are calculated using the method of moments within kinetic models, as in the prior art.
For a given property s, the property distribution may be described by a frequency function f_{s} when the property is a discrete variable, and by a density function f(s) when the property s is continuous. Therefore, f_{s} and f(s) represent the portion (e.g., number, weight, volume, fraction) of the population whose property is exactly s (discrete) or whose property lies between s and s+ds. The frequency and density distribution functions are respectively ##EQU3## with s_{0} being the initial value of s, and S an arbitrary higher value (Biesenberger, 1983).
Distribution moments may be defined from the origin of the average property, i.e., property is equal to 0, or from the mean value of that property. The moments employed in the preferred embodiment use the first approach. In this case, the generalized form of the relationship between distribution moment and distribution function is as shown in the following equation. ##EQU4## where μ is the moment;
k is the moment order (e.g., 03 for zeroth through third moment);
s is the property value (e.g., chain length, molecular weight, particle size, etc.);
f_{s} is the frequency function; and
f(s) is the density function.
The average properties are calculated as ratios of the moments. Number average is the ratio of first to zeroth moment, μ_{1} /μ_{0}. Weight or Volume average is the ratio of second to first moment, μ_{2} /μ_{1}. Zaverage is the ratio of third to second moment, μ_{3} /μ_{2}.
For the case of chain length distribution the moments may be calculated using
λ.sub.m =Σn.sup.m Q.sub.n
where
λ is the moment;
m is the moment order;
n is the chain length or degree of polymerization; and
Q_{n} is the number of moles of polymer of length n (when Q_{n} is normalized, the moments λ and μ are the same).
The average chain length properties are then
DPn=λ_{1} /λ_{0}
DPw=λ_{2} /λ_{1}
DPz=λ_{3} /λ_{2}
PDI=λ_{2} /λ_{0}
A similar definition of moments for the frequency distribution is applied to molecular weight. In the present invention, it is applied to chain length, then the average molecular weight values are determined using the average degree of polymerization and average segment molecular weight.
Method of Instantaneous Properties
The present invention is directed to polymer processes involving chain growth polymerization (other than true living), for any active center type (radical, anionic, cationic, ZieglerNatta, metallocene (coordination and otherwise)), for any process type (solution, slurry, gas phase, suspension, emulsion, etc.), for any reactor operation type (batch, semibatch, continuous), for any reactor type (CSTR, PFTR and combinations thereof). Briefly, pulses of dead polymer are continuously generated in the polymerizing mass. These pulses have a mass and molecular properties (such as distributions of composition, molecular weight, long chain branching and stereoregularity, etc.). The present invention instantaneous property method uses this information and provides effective methodologies (with high information transfer efficiency) for (1) calculation of the distributions of composition, molecular weight, long chain branching and stereoregularity, for the dead polymer in each pulse; and (2) tracking the mass and distributions of each pulse of polymer through the transient and steadystate manufacturing system (which includes polymer reactors and ancillary equipment (distillation towers, dryers, extruders, and the like).
Applying the method of moments for the calculation of property distributions has several drawbacks as previously mentioned. In addition to CPU requirements and computational complexity, a larger number of moments than currently calculated in process modeling programs/systems would be required. A knowledge of even a larger number of moments of a distribution does not permit one to unambiguously construct a complex distribution. One must therefore look beyond the method of moments for a more powerful method to predict these complex distributions for a polymer manufacturing system.
A better approach for generating distributions (of molecular weight, composition, stereoregularity or long chain branching) consists of (i) calculating or using existing distribution functions which are valid solutions of the population balance equations (such as the SchulzFlory molecular weight distribution or the Stockmayer bivariate distribution of composition and molecular weight) for polymer chains produced "instantaneously", (ii) storing the kinetic parameters of these distributions throughout the simulation calculations for the network of equipment used for polymer manufacture (polymer reactors, reactive extruder, separators, etc.), and (iii) later using these kinetic parameters and their associated distribution functions and polymer mass or polymer mass production rates to construct the full distribution (composition, molecular weight, stereoregularity, long chain branching) of polymer accumulated in the reactor system. Such an approach has its beginnings in Hamielec, A. E., "Polymerization Processes," Ullmann's Encyclopedia of Industrial Chemistry, 5th ed. Vol. 21, pp. 324330 (1992).
In the simplest case, linear polymerization in a single CSTR reactor, the ratios of termination and chain transfer reaction rates to propagation reaction rates are stored. The instantaneous chain length distribution is expressed as a function of these ratios and chain length. For the case of two CSTRs in series, at steadystate, the product polymer weight chain length distribution function is the weighted average of the distribution function in each CSTR taken separately. The case of a plug flow reactor may be approximated using multiple CSTR's, and similarly for a batch reactor.
By looking at the treatment of such reactor configurations, applicants deduce that the final polymer distribution is a result of the entire system of reactors. For this reason, the molecular weight distribution (MWD) modeling implementation in the present invention takes into consideration the proper data structure to track distribution parameters at every point in a process flowsheet. Users are able to request MWD from any point in the flow sheet, and from this point the present invention software flowsheet connectivity information may be used to track polymerization history. The solution of the population balance equations to generate distribution functions for polymer chains produced "instantaneously" will now be described in detail.
The calculation of chain length distribution for a batch reactor from reaction rate parameters for linear addition polymerization was described by Hamielec (Hamielec, 1992). Consider the equations for the generation and consumption of free radicals. A similar approach may be used for other active centers (ZieglerNatta, metallocene, etc.) ##EQU5## where [R_{I} ]=2K_{d} f[I] is the initiation rate.
Let us introduce two dimensionless parameters τ and β. ##EQU6## where R_{p} =K_{p} [R^{0} ][M] is the propagation rate;
R_{td} =K_{td} [R^{0} ]^{2} is the rate of termination by disproportionation;
R_{tc} =K_{tc} [R^{0} ]^{2} is the rate of termination by combination; and
R_{f} =K_{fm} [R^{0} ][M]+K_{fT} [R^{0} ][T] is the total rate of all chain transfer reactions.
If we assume that the stationarystate hypothesis holds, then the initiation rate is equal to the sum of the termination rates, R_{I} =R_{td} +R_{tc}.
The equations for the rate of generation and consumption of radicals are written as follows: ##EQU7## Therefore
[R.sub.r.sup.0 ]=[R.sup.0 ](τ+β)Φ.sup.r Equation 1.7
where ##EQU8## The rate of production of polymer molecules of chain length r, R_{Fp} (r) is given by the following equation. ##EQU9## Substituting [R_{r} ^{0} ] from Equation 1.7 in Equation 1.9 gives ##EQU10##
Therefore, the instantaneous weight chain length distribution may be calculated from rate of production of polymer molecules as follows. ##EQU11##
In other words, W(r) is the weight chain length distribution of dead polymer chains produced in a small time interval t to t+dt, in a batch reactor. W(r) is also the weight chain length distribution of dead polymer chains produced in a CSTR operating at steadystate.
If βτ, which is the case when the polymer chains are formed by chain transfer or by termination by disproportionation, this equation reduces to ##EQU12##
In this equation, 1/(1+τ) is the probability of growth for a polymer radical, and τ/(1+τ) is the probability that a polymer radical stops growing. Since r is usually large, W(r) in Equation 1.11 can be approximated as a continuous function with small error. ##EQU13## For most freeradical polymerizations (τ+β)1 and is of the order 10^{6} to 10^{2}.
The weightaverage chain length for polymer produced instantaneously is given by ##EQU14##
The instantaneous numberaverage chain length is given by ##EQU15##
The polydispersity index for polymer produced instantaneously is given by ##EQU16##
Equation (1.13) applies to both homo and copolymerization with two or more monomer types. When chain growth polymerizations are done with active center types other than radicals (ZieglerNatta, metallocene, etc.) β=0 in Equation (1.13), and the instantaneous chain length distribution becomes a single parameter τ distribution, which is in fact Flory's most probable distribution with a polydispersity index of 2.0.
Equation (1.13) is the main expression used in the preferred embodiment to generate chain length distribution of linear homopolymer and copolymer chains. Within the context of a polymerization reactor, this expression is valid for the case of linear chains of a homopolymer or copolymer produced in a single CSTR at steadystate. For the case of two CSTR's in series, the end product polymer distribution is a composite which is a weighted average of the distributions of polymer produced in the first and the second reactor. ##EQU17## where m=m_{1} +m_{2} is the total mass of polymer produced in the first and the second reactor per unit time. The distribution function in each reactor is given by Equation (1.13) with the τ and β, varying from reactor 1 to reactor 2, and independent of time under steadystate operation.
In the present invention, a plug flow reactor is divided into several volume elements and treated as a series of CSTR's. The τ, β, and polymer mass production rate values are stored for each volume element and later used for the calculation of the composite chain length distribution function. A batch reactor is handled using a similar approach. In this case, the τ, β, and polymer mass values are stored for each time increment.
For linear chains of a copolymer, the difference from the homopolymer case is only in the calculation of the reaction rates for propagation, termination, and transfer reactions, R_{p}, R_{tc}, R_{td}, and R_{fm}.
Mechanism for Tracking Distributions:
In the present invention, the method of instantaneous properties is used to generate chain length and other property distributions in contrast to the prior art method of moments of distributions used to generate average properties of the polymer. The present invention method is applied at two levels:
the reactor level for determining the mass or mass production rate and distribution (composition, molecular weight, stereoregularity and/or long chain branching) of polymer newly produced within the vessel (local distribution), and
the flowstream level for determining how the mass of polymer produced "instantaneously" in a reactor upstream is distributed in streams downstream at some later time and to what extent the distribution of polymer molecular properties (composition, molecular weight, stereoregularity and long chain branching) is distorted due to physical and chemical changes downstream (cumulative distribution).
Within polymer reactors, the present invention method of instantaneous properties is used directly to determine the distribution of newly produced polymer. As kinetic calculations are being performed, the values for the instantaneous kinetic parameters τ and β, as calculated using Equations 1.3 and 1.4 respectively within the kinetic model, are saved for later calculations of the distribution. In addition, the polymer mass or polymer mass of production rate corresponding to these values is saved.
For a CSTR reactor operating at steadystate, a single value of τ and β is needed. For a plugflow or batch reactor, a value of τ and β is needed at every profile point. In the case of the plugflow reactor, the profile points represent equalsized axial sections along the reactor length. In the case of the batch reactor, profile points are stored in fixed time intervals.
To calculate the distribution of polymer produced at the exit of the reactor, Equation 1.13 is applied with the calculated τ and β for the CSTR reactor. For a plugflow or batch reactor each profile element is represented as a CSTR for which that same equation is applied. To determine the distribution at the exit of these reactors, Equation 1.17 is applied to the total number of profile elements. This local distribution obtained is combined with distribution from the reactor inlet, where applicable and using Equation 1.17, then transferred through the outlet to downstream unit operations. In the preferred embodiment, the distribution is available to the end user both as a data table and as a graphic plot.
For multisite kinetic models, such as required for ZieglerNatta catalysts, an additional dimension is introduced. The τ and β parameters are stored for each site type. Therefore, the number of parameters for the case of a CSTR will be the userspecified total number of site types. The distribution produced on each site type is calculated using Equation 1.13, then Equation 1.17 is appropriately modified to account for multisite types (as discussed later) to calculate the composite polymer distribution. This procedure is applied once more to represent plugflow and batch reactors as multiple CSTR reactors.
It is noted that in the preferred embodiment, in any given reactor, the complete flowsheet connectivity is not needed in order to calculate the distribution of accumulated polymer. Each inlet stream has associated with it the distribution of accumulated polymer up to that point in the flowsheet.
Distributions in Process Streams
The polymer distribution calculated within polymer reactors is transferred into the outlet stream. This allows flowsheeting of the cumulative distribution, i.e., the data follows the polymer component throughout the flowsheet. The cumulative distribution is stored within the stream as a Property Set or PROPSET in the preferred embodiment. In the preferred embodiment, the process modeling software provides several different vehicles for associating data with process streams. The first is the stream vector which contains composition and state information. Also there are component attributes which are employed in the preferred embodiment. Another vehicle is the PROPSET.
One basic difference between PROPSETS and the other two categories of stream data structures is the fact that the latter are processed during convergence calculations while PROPSETS are not. If distributions were processed during convergence, there would be a significant computing penalty due to the large amount of data they introduce. The information used for calculating the distributions is derived from converged quantities. Thus, there is no need for applying convergence calculations to the distribution data itself.
Another difference is the fact that PROPSETS are not part of the default stream structure available to model analysis tools. Endusers often access the standard stream structure either through model analysis tools or in code in user FORTRAN subroutines. The instantaneous polymer distribution data needs to be accessed differently than other stream data. In the preferred embodiment, this polymer distribution data is carried in nonaccessible storage space or similar memory.
FIG. 2 illustrates the procedure followed to generate the property distributions, for steadystate operation, in the present invention.
Referring to FIG. 2, the preferred embodiment software procedure 21 begins with user input step 23. A user interface supports user input step 23 and prompts the enduser to enter indications describing (i) the polymer property distribution desired (e.g., molecular weight distribution, composition distribution, long chain branching, stereoregularity and the like), (ii) the subject chemical process/reaction desired to be simulated (including reaction model and specific kinetic parameters), and (iii) the subject manufacturing system including a network of reactors (e.g., one or more kinetic reactors: CSTR, plugflow, or other reactors, operating in batch, semibatch or continuous operation, and the like). In particular, for the subject reactor network, the enduser is prompted to specify reactor configuration and operating conditions. In response, procedure 21 initializes a flow sheet which maps product flow and reactor configuration.
Further in response to the indications entered by the enduser in step 23, procedure 21 generates data elements to provide working distribution information to subsequent steps. Specifically, procedure 21 generates (i) a distribution identifier, (ii) a flag or other indicator for indicating distribution type, (iii) a locator (referred to as a stream identifier) indicating location in the flow sheet, and (iv) parameters τ, β and m for feed streams containing polymers. These data elements are passed to an input processing step 25.
Input processing step 25 receives and is responsive to the passed data elements. In particular, input processing step 25 processes distribution characterization data and sets up a distribution bead or vector for storing distribution data. Included in the distribution bead set up, the input processing step 25 initializes an appropriate, fixed amount of storage (memory). Further in response to the reactor and reaction specifications given by the enduser in input step 23, input processing step 25 initializes a reaction data block. This includes allocating in the reaction data block space for each set of τ_{i}, β_{i} and m_{i}. A dimension parameter is set to 1 for CSTR unless the enduser requests residence time distribution in step 23. Input processing step 25 ties the dimension parameter to profile points for plug flow and batch reactors.
Next, calculation step 27 performs reactor calculations using the equations previously discussed. The results of each calculation are stored in the reaction data block. Also kinetic calculations are performed on the stored data. τ_{i}, β_{i} and m_{i} result, and for each i (each set), τ_{i}, β_{i} and m_{i} are associated with a respective profile point. Other distribution parameters are also associated with respective profile points. The association of the various distribution parameters per profile point enable or effect calculation of the full distribution of target properties at the end of the reaction. After calculating full property distributions, calculation step 27 stores the same in the distribution bead upon exit from the reactor.
A report generator 29 is coupled to receive characterization data from input processing step 25 and the full property distributions as calculated by calculation step 27. More accurately, report generator 29 retrieves from the distribution bead, the property distributions for the polymer product and therefrom provides display of distributions of property sets (e.g., molecular weight distributions, composition distributions and the like) to the enduser upon user request. Also the report generator 29 enables distribution data to be displayed in tabular format and/or graphically upon user request. Using graphic technology common in the art, reporter 29 cooperates with a graphical user interface which enables enduser interaction with the various displays (e.g., from tables to graphs, etc.) as viewed on a monitor 19.
Accordingly, procedure 21 provides modeling of polymer reactions in reactor networks by tracking instantaneous property measures (reaction rates) and hence instantaneous property distributions throughout the reactor network and by calculating full property distributions upon exit of the reactor network.
It is understood that the foregoing software procedure/routine 21 is executed in a digital processor of a computer system, such as that illustrated in FIG. 3. Briefly, computer system 13 is formed of (a) a digital processor 15 with accompanying working memory 17, disk memory 31 and the like, and (b) I/O peripherals common in the art including, but not limited to, a viewing monitor 19, keyboard 33 and mouse 35. Also digital processor 15 may be a node or server in a network of computers and is, for example, of the Intel xx486 processing capabilities or better.
Preferably routine 21 is executed in working memory 17 of digital processor 15 with user input being received from I/O peripherals 33, 35 and visual output being provided on monitor 19. In the preferred embodiment, routine 21 is part of a multiplicity of cooperating (or compatible) software modules 37 for modeling/simulating, monitoring and otherwise analyzing reactor systems (including reactor networks and chemical processes/reactions carried out on the reactor network). Examples of such software modules are Aspen Plus and Speedup (both products of assignee of the present invention). To the extent necessary for understanding the present invention, these or similar additional software modules are disclosed in U.S. Pat. Nos. 5,008,810 and 5,666,297 herein incorporated by reference.
It is understood that other computer configurations and other software modules/architecture are suitable and in the purview of one skilled in the art. For example, a database of reactor system information/data may be involved. The database may reside locally in memory 17 or off disk 31 or the like. The various software modules 37, 21 may share the database information for respective processing. The foregoing description in FIGS. 2 and 3 are for purposes of illustration and not limitation of the present invention.
Restating the foregoing, the present invention introduces novel modeling methods and apparatus for steadystate and dynamic polymer reactor simulations, which have high information transfer efficiencies and permit the efficient calculation of the full distributions of composition, molecular weight and long chain branching, for linear and branched homo and copolymers having any number of branch points and distinct monomer types. These novel modeling methods/apparatus are next illustrated using examples of chaingrowth polymerization involving freeradical polymerization and coordination polymerization with catalyst systems (ZieglerNatta and metallocene catalyst systems, for example) having single and multiple active site types.
To illustrate the present invention methodology of efficiently tracking (high efficiency of information transfer) distributions of composition, molecular weight and long chain branching through transient and steadystate polymer reactor networks, consider a polymerization reactor which is well agitated so that spatial variations in temperature and concentrations of reactants and products are negligible. For solution or bulk polymerizations, this implies that temperature and concentrations of reactants and products dissolved in a single phase are spatially uniform throughout the reacting mass in the reactor. Of course, volatile solvents, reactants and small molecule byproducts may partition in the gaseous headspace of the reactor, but this partitioning can easily be accounted for. For emulsion, suspension, dispersion and gas phase slurry polymerization processes, spatial uniformity of concentrations of reactants, solvents, polymer product and others is more difficult to define.
Now during polymerization, reactions may take place in more than one phase and there is clearly a discontinuity in concentrations at the interface of phases. For example, in emulsion polymerization for a wellmixed reactor, concentrations in the aqueous phase are considered spatially uniform. However, in the polymer phase (or in the polymer particles) a smaller particle may contain fewer radicals that a larger polymer particle and the distributions of composition, molecular weight and long chain branching of the polymer in these two sized particles may differ significantly. Applicants therefore define wellmixed for multiphase polymerizations in the following manner.
Choose a volume element, containing a large number of polymer particles, which is much smaller than the total volume of the reacting mass (volume of all phases, in the reactor). The polymer particles are coagulated in the volume element and are mixed and analyzed as a whole. If the distributions of composition, molecular weight and long chain branching are essentially the same for all volume elements, Applicants then consider the contents of the reactor to be wellmixed. This wellmixed or wellagitated reactor can be operated in batch, semibatch and continuous mode. Batch and semibatch operations are inherently transient while continuous operation may be steadystate as well as transient.
Batch and SemiBatch Operation
The present invention methodology is first described qualitatively, introducing certain fundamental ideas. A batch and semibatch operation under the present invention methodology are considered together and are followed by a consideration of the transient continuous operation.
Consider, for example, a wellmixed polymer reactor operating as a batch or semibatch reactor. When in semibatch operation, small molecule ingredients are added to the reactor over time with some timevarying feed rate. With batch operation, all of the small molecule ingredients and possibly some polymer heel is added to the reactor at time t=0 over a small time interval. The temperature of the polymerizing mass in the reactor may vary over the total polymerization time t_{F}. From time t=0 to t=t_{F} (total polymerization time), dead polymer chains with distributions of composition, molecular weight and long chain branching are produced continuously. The rate of dead polymer production and the distributions of composition, molecular weight and long chain branching may all vary with time for dead polymer produced instantaneously (i.e., dead polymer chains produced in a "small" time interval, much smaller than the total polymerization time) as well as for dead polymer accumulated over the time interval, 0<t≦t_{F}, where t represents some time during the time interval, 0 to t_{F} where t_{F} is the total polymerization time.
Let us now introduce some additional symbols:
______________________________________W(t,t) = weight of dead polymer produced (or born) at time t = t in a small time interval Δt, where t is the birthtime of "instantaneous" dead polymer, with t ≦ t.W(t,t) ≦ W(t,t) in general, and is at a time later than t. However, if there are no long chain branching reactions (e.g., chain transfer to polymer, terminal branching or addition to pendant double bonds), then W(t,t) = W(t,t) and thus the weight of instantaneous dead polymer and its distributions of composition and molecular weight remain constant in the time interval, 0 to t.sub.F. However, because the weight and distributions of composition and molecular weight of instantaneous dead linear polymer chains may change with birthtime, t, the distributions of composition and mass for the accumulated dead polymer (accumulated in the time interval, 0 to t) may change with time t.W(r, i, t, t) is the weight chain length distribution of instantaneous dead polymer born at time t at present time t with a chain length r, and having i long chain branches per polymer molecule. So r is the number of monomer units and i is an integer in the set {1,2, . . . 50} with different number of long chain branches per molecule defining a family of branched chains. For the case of linear chains with no long chain branching reactions, i = 0 andW(r, 0, t, t) = W(r, 0, t, t) which means that the chain length distribution of dead polymer born at time t does not change with time t.______________________________________
However, it should be mentioned that W(r, 0 t, t) may change with birthtime t, as variables such as temperature and concentrations of reactants, may in general change with time.
The weight chain length distribution of the dead polymer accumulated in the time interval 0 to t is given by ##EQU18## where W_{p} is the mass of dead polymer accumulated in the time interval, 0 to t, and dw_{p} is the mass of instantaneous dead polymer produced in the time interval, t to t+dt. Where W(r, 0, t, t) is the weight fraction of polymer with chain length r produced at time t. R_{p} (t) is the rate of propagation at time t. V(t) is the volume of the polymerizing mass at time t, and MW is molecular weight of the monomer repeat unit
W(r, 0, t, t) may be derived analytically, using the differential form of the population balance equations for dead polymer and the algebraic form of the population balance equations for live polymer after application of the stationarystate hypothesis (See Archie E. Hamielec and Hidetaka Tobita, "Polymerization Processes," pp. 324326, Ullmann's Encyclopedia of Industrial Chemistry, Vol. A21, VCH Publishers, Inc. (1992)).
Equation A1 may be written in the form of an ordinary differential equation if desired, as follows: ##EQU19## and in a batch reactor ##EQU20## Therefore, ##EQU21## where K_{p} is the propagation constant;
M is the concentration of monomer;
Y_{0} is the concentration of polymeric radicals; and
V is the volume of the polymerizing mass.
Thus the weight chain length distribution of dead polymers produced instantaneously is ##EQU22## where the two dimensionless parameters, τ(t) and β(t), which in general depend on birthtime t of the instantaneous dead polymer, are given by:
Freeradical polymerization
τ(t)=(R.sub.trm +R.sub.trCTA +R.sub.trs +R.sub.td)/R.sub.p
β(t)=R.sub.tc /R.sub.p
where
R_{trm} is the rate of chain transfer to monomer;
R_{trcta} is the rate of chain transfer to chain transfer agent (CTA);
R_{trs} is the rate of chain transfer to solvent;
R_{td} is the rate of termination by disproportionation;
R_{tc} is the rate of termination by combination; and
R_{p} is the rate of propagation.
All of these rates may vary with time. These rates are readily available in existing software involving freeradical polymerization.
Polymerization Using Coordination Catalyst Systems
These coordination catalyst systems may be singlesite type or multiplesite type. Applicants treat these separately as follows:
Singlesite type catalyst systems
For these systems β(t)=0, and τ(t) is given by
τ(t)=(R.sub.β +R.sub.trm +R.sub.trH2 +R.sub.deac)/Rp
where
R.sub.β is the rate of βhybride elimination;
R_{trH2} is the rate of chain transfer to hydrogen; and
R_{deac} is the rate of deactivation.
Again, these rates are readily available in software for polymerization using coordination catalyst systems and W(r,0,t,t)=τ(t)^{2} r*exp (τ(t)r) or Flory's mostprobable distribution. The weight chain length distribution of the accumulated dead polymer may be calculated as discussed above in Equation A1 with freeradical polymerization.
Multiplesite type catalyst systems
To apply the novel methodology to multisite type catalysts, one must characterize the catalyst systems using polymerization followed by polymer characterization (GPC and TREF) followed by deconvolution (See Archie E. Hamielec and Joao B. P. Soares, "Polymerization Reaction EngineeringMetallocene Catalysts," Prog. Polym. Sci., Vol. 21, pp. 651706 (1966)). This will provide the relative propagation rates for the different site types as well as the values of τ(t) for each site type. The instantaneous weightchain length distribution for a multisite type catalyst system is given by ##EQU23## where N is the number of site types j and W_{j} (t) is the weight fraction of polymer produced on site type j at birthtime t. The calculation of the weight chain length distribution of the polymer accumulated in the time interval 0 to t is as stated above for freeradical polymerization in Equation A1.
Treatment of copolymerization.
For copolymerizations with two or more monomer types, the population balance equations may be reduced to those for a homopolymerization with the introduction of appropriate pseudokinetic rate constants (refer to H. Tobita and A. E. Hamielec, Polymer 32, 2641 (1991)). In other words, Equation (A) the instantaneous chain length distribution applies for multicomponent polymerizations as well as homopolymerizations.
As far as the chemical composition distribution is concerned, Stockmayer's bivariate distribution (See W. H. Stockmayer, J. Chem. Phys., Vol. 13, p. 199 (1945)) which applies only for binary copolymerizations, covers chain length and composition distributions and is given by ##EQU24## where W(r, y, 0, t, t) is the bivariate distribution of chain length r and composition deviation y (deviation from the average mole fraction of monomer 1 in the copolymer) for linear dead binary copolymer chains produced instantaneously at birthtime t in a small time interval, Δt.
∇(t)=F.sub.1 (t) (1F.sub.1 (t)) K(t),
K(t)=[1+4F.sub.1 (t) (1F.sub.1 (t)) (r.sub.1 r.sub.2 1)].sup.0.5, ##EQU25## F.sub.1 (t) is the mean mole fraction of monomer 1 in the copolymer produced instantaneously at birthtime t,
r_{1} and r_{2} are reactivity ratios in the context of the terminal copolymerization model, and
MW_{1} and MW_{2} are the respective molecular weights of monomer 1 and 2 repeat units in the binary copolymer chains.
Stockmayer's bivariate distribution (Equation B) may be applied to a binary (e.g., 2 monomer types) freeradical copolymerization or to a binary copolymerization with a singlesite type coordination catalyst system as is. For a multisite type coordination catalyst, Stockmayer's bivariate distribution may be applied to each site type, but having different parameters. For example, different site types will have a different pair of reactivity ratios. The use of Stockmayer's bivariate distribution as the bivariate distribution of binary copolymer produced instantaneously is generally available (i.e., calculated by existing art techniques).
Unfortunately, an analytical trivariate distribution which accounts for three monomer types has yet to be derived analytically. However, if chain lengths are sufficiently large, the variance of the composition distribution part of Stockmayer's bivariate distribution approaches zero and one may assume with small error that copolymer chains produced instantaneously all have the same mole fractions of monomer types. This, in turn, may be applied to, say, ternary or higher copolymerizations, and would permit one to readily calculate a trivariate or higher distribution for the accumulated copolymer using Equation A for the weight chain length distribution part.
Polymer Chains with Long Chain Branching.
We are still considering batch and semibatch operation; however, now long chain branching reactions (chain transfer to polymer, terminal branching (i.e., terminal double bonds at ends of dead polymers) and addition to pendant double bonds for freeradical processes, and terminal branching and addition to pendant double bonds for coordination polymerization) and their effect on the tracking methodology is now considered. With these long chain branching reactions, dead polymer chains may be reactivated to give live polymer chains connected chemically to an active center or site for propagation. In other words, W(t,t), the weight of dead polymer produced instantaneously at time t will decrease in the time interval t>t, and weight distribution of the instantaneous dead polymer W(r, i, t, t) may or may not be distorted by the long chain branching reactions.
Terminal branching rate is proportional to the concentration of dead polymer chains and as a consequence will not distort the molecular weight distribution of dead polymer chains having one terminal double bond. The mass of these dead polymer chains will decrease, however. Transfer to polymer and addition to pendant double bonds reactions will distort the molecular weight distribution because these rates are greater for longer chains which have either more extractable backbone atoms or more pendant carboncarbon double bonds per chain. To illustrate these ideas, we now consider chain transfer to polymer.
Chain Transfer to Polymer.
Suppose that W(r, i, t, t) is the weight chain length distribution of dead polymer formed instantaneously at time t, having i≧0 long chain branches per polymer molecule. Chains in W(r, i, t, t) are consumed, and the distribution distorted by chain transfer to polymer in the time interval t>t as follows.
Consider a batch reactor ##EQU26## where P(r, i, t, t) is the concentration (in moles/dm^{3}) of dead polymer having chain length r, i trifunctional long chain branches per polymer molecule which was born at time t at present time t. K_{trp} is the chain transfer to polymer rate constant in dm^{3} /(mole) (min). Y_{0} is the total polymeric radical concentration in moles/dm^{3}. Equation C may be converted to ##EQU27## and solved analytically for isothermal polymerization to give: ##EQU28## where ##EQU29## and K_{p} is a propagation constant. Equation D may be used to calculate the distortion experienced by dead polymer chains after birth in the time interval t to t. Note that N_{M} is the number of moles of monomer in the batch reactor at any time t, and N_{M} =N_{M} at time t=t.
The calculation of W(r, i, t, t) using Equation D is then as follows. Equation D may be used to calculate the change in the concentration of dead polymer of chain length r and having i long chain branches per molecule produced "instantaneously" at time t later in the time interval, t>t and thus the distortion in the distribution due to chain transfer to polymer.
Further, the use of the invention instantaneous property method to calculate W(r,i,t,t) does not give an analytical solution for branched families with i≧1. The family with linear chains with i=0 has an analytical solution. The frequency distribution F(r,i,t,t) for, i=0, is given by
F(r,0t,t)=(A/B)η.sup.2 exp (ηr)
where
A=(τ(t)+γ(t))+(β(t)/2) (r1) (τ(t)+β(t))
B=η(τ(t)+γ(t))+(β(t)/2) (τ(t)+β(t)),
η=τ(t)+β(t)+γ(t)
τ(t) and β(t) have already been defined. ##EQU30## where K_{trp} is the chain transfer to polymer rate constant
Q_{1} is the first moment of all the accumulated dead polymer chains at t=t.
When
γ(t)=0 i.e., no chain transfer to polymer occurs, F(r,0,t,t) is the associated frequency distribution of weight chain length distribution given by Equation A. The weight chain length distribution may be calculated from the frequency distribution of chain length as follows: ##EQU31## In general, the frequency distribution may be calculated using p(r,i,t,t), the concentration of dead polymer of chain length r and having i trifunctional long chain branch points per polymer molecule as follows: ##EQU32## When the invention instantaneous property method does not provide an analytical solution for the population balance equations (such as the SchulzFlory distribution and Stockmayer's bivariate distribution) it is recommended that the invention instantaneous property method be used to calculate the leading moments for the distribution for dead polymer produced instantaneously and that these moments be tracked through transient and steadystate polymer manufacturing systems. When the full distributions are required at any point in the manufacturing system, a 2parameter empirical distribution function (such as the SchulzZimm distribution) is then used, and the leading moments are used to calculate the two parameters (e.g., τ and β or the Schulz/Zimm parameters) for dead polymer produced instantaneously. Otherwise, calculation of the full distributions numerically for i≧1 would require excessive data storage and computation time.
The weight chain length distribution of the polymer accumulated in a batch reactor for families having i long trifunctional branches per polymer molecule is found by integrating Equation D with respect to N_{M} in the interval 0 to N_{M} which is equivalent to the time interval 0 to t. When temperature of polymerization is changing with time, however, Equation D should be used piecemeal over a series of short constant temperature intervals. C_{p} would have a different value for each temperature level. Equation D may also be used to track the leading moments of the distributions for dead polymer produced instantaneously (i.e., polymer pulse). At any point in time, these leading moments may be used to estimate the 2parameters of say the SchulzZimm distribution, and in turn the SchulzZimm distribution may be used to solve the moment closure problem.
We now consider continuous operation under transient conditions with steadystate operation as a special case and to illustrate the basic ideas, one CSTR (continuous stirred tank reactor) is considered which is being operated full or with a reacting volume which does not change with time. In other words, we are not considering reactor filling or reactor drainage, which are special cases, for the present discussion.
Referring to FIG. 1, v_{0} is the total volumetric flowrate of all ingredients into the reactor 11 in dm^{3} /minute. v is the total volumetric flow rate of polymer, unreacted monomers, solvent and other ingredients out of the reactor 11 in dm^{3} /min. Generally v<v_{0}. V is the volume of polymerization mass in the reactor 11 in dm^{3}, and may be (i) a homogeneous solution of polymer in monomers and solvent, or (ii) a suspension of polymer particles in a continuous liquid or gaseous phase, which may be an organic or water.
During reactor operation, dead polymer chains are being generated continuously and uniformly throughout the polymerizing volume in the reactor 11. Neglect long chain branching reactions for now, and consider the production of linear homopolymer chains. With continuous operation and outflow, the series of pulses of dead polymer which are formed over a period of time will lose some of their mass due to outflow but their weight chain length distributions will not change due to outflow. W(t,t), the mass of instantaneous dead polymer (or a pulse of dead polymer) born at time t is calculated for the time interval t>t as follows: ##EQU33## where θ=V/v is the mean residence time of the vessel which is assumed to remain constant during transient operation of the CSTR. This permits an analytical solution of Equation E as follows
W(t, t)=W(t, t) exp ((tt)/θ) Equation F
If it is necessary to account for the variation of the mean residence time with time (e.g., output flow rate decreases), Equation E may readily be solved numerically. It should also be mentioned that for transient operation of the CSTR, W(t, t), τ(t) and β(t) will in general change with birthtime of the series of pulses of dead polymer. It should also be mentioned that outflow does not distort or change W(r, 0, t,t) in any way, and for this example with linear polymer chains, W(r, 0, t, t)=W(r, 0, t, t). Treating W(t, t) as a continuous variable, the weight chain length distribution of dead polymer accumulated in the CSTR at time t is given by: ##EQU34## At steadystate, W(r,0,t,t) is independent of birthtime, t and the weight chain length distribution of dead polymer in the CSTR is equal to W(r,0,t,t). In general for linear homo or copolymer chains ##EQU35## For ionic (coordination or otherwise) catalyst systems, set β(t)=0.
For the more general case, where the mean residence time of the CSTR changes with time, Equation G may be written as ##EQU36## where W_{p} (t), the total mass of polymer accumulated in the CSTR at time t, is a quantity which is readily available in existing process modeling software for the simulation of polymer reactors.
Let us now consider 2CSTR's of equal volume in series under transient operation. In the first instance, we neglect volume changes when monomers are converted to polymer and assume that the reactors operate with constant fillage. For two CSTR's of equal volume in series ##EQU37## where W_{12} (t,t) is the weight of the pulse of dead polymer formed at time t in CSTR(1) which is now in CSTR(2) at some later time t. ##EQU38## where W_{12} (t) is the total weight of all pulses of dead polymer formed in CSTR(1) in the time interval 0 to t, which is now in CSTR(2) at time t.
W.sub.22 (t,t)+(W.sub.22 (t,t) exp ((tt)/θ) Equation K
where W_{22} (t, t) is the weight of the pulse of dead polymer formed at time t in CSTR(2) which is now in CSTR(2) at some later time t.
To calculate the weight chain length distribution of dead polymer in the exit stream of CSTR(2), we must define τ's and β's, the dimensionless parameters in Equation A, separately for each CSTR as τ_{1} (t), τ_{2} (t), β_{1} (t), β_{2} (t). It may be convenient to first calculate W_{12} (r, 0, t), the weight chain length distribution of dead polymer formed in the time interval, 0 to t, in CSTR(1) and now flowing out of CSTR(2) at time t, and then W_{22} (r, 0, t), the weight chain length distribution of dead polymer formed in the time interval 0 to t in CSTR(2) and now flowing out of CSTR(2) at time t. W_{12} (r, 0, t) and W_{22} (r, 0, t) may be later combined to calculate the overall weight chain length distribution in the outlet stream of CSTR(2).
The treatment of any number of CSTR's in series is similar to the above example of two CSTR's in series and straightforward, for linear homopolymer and copolymer chains when the CSTR train is operated in transient mode with reactor reaction volumes constant with time. For CSTR fillage and drainage, a modified procedure must be employed as follows:
Example for reactor fillage: Suppose that we have a single CSTR which is empty at time t=0 when the inlet flows are set to their steadystate values. Let t_{fill} be the time required for the reactor to fill and for species to begin flowing out. For the time interval 0 to t_{fill} applicants consider the operation to be semibatch and use appropriate procedures to calculate the total weight of polymer and its weight chain length distribution at time t=t_{fill}. The weight of this polymer remaining in the CSTR, in the time interval t≧t_{fill} is given by: ##EQU39## W(t_{fill}, t_{fill}) is the weight of dead polymer in the CSTR when t=t_{fill}. For t≧t_{fill} the weight chain length distribution of this dead polymer accumulated in the CSTR in the time interval 0 to t_{fill}, W(r, 0, t_{fill}) can be readily calculated using procedures already discussed above. W(r,0,t,t), where t≧t_{fill}, can easily be calculated using procedures already described above for a single CSTR in transient operation.
Next are examples for a train of CSTR reactors operating at steadystate involving the production of linear homopolymer and copolymer chains.
Freeradical polymerization: Consider NCSTR reactors in series operating at steady state
W(r, 0)=(m.sub.1 W.sub.1 (r,0)+m.sub.2 W.sub.2 (r,0)+ . . . +m.sub.n W.sub.N (r,0))/m Equation M
where W(r,o) is the weightchain length distribution of the dead polymer (a composite of dead polymer produced in each of the NCSTR) flowing out of CSTR(N), and m_{N} is the production rate of dead polymer in (gm/unit of time) in CSTR(N). For solving Equation M, total production rate of dead polymer m is readily available in existing process modelling software. That is, existing process modeling software calculates a total or sum of production rates of dead polymer m as
m=m.sub.1 +m.sub.2 + . . . +m.sub.N Equation N
The weight chain length distribution of dead polymer formed in reactor N is given by ##EQU40## Coordination multisite type catalyst system.
Let's now calculate the instantaneous distribution as a composite of instantaneous distributions produced by each site type. Let m_{ij} be the production rate of dead polymer on site type i in reactor j. Then ##EQU41## where W_{ij} (r,0) is the weight chain length distribution of dead polymer produced instantaneously on site type i in CSTR j, and ##EQU42## where NS is the number of site types, and where NR is the number of CSTR reactors.
Next consider a single CSTR operating at steadystate with long chain branching reactions in effect. With long chain branching reactions, it is appropriate and convenient to consider the weight chain length distributions of different polymer families having 0, 1, 2, 3, etc. long chain branches per molecule. Applicants consider chain transfer to polymer as the sole reaction producing long chain branching. Instantaneous distributions for dead polymer having 0, 1, 2, 3, etc. long chain branches will be independent of birthtime for steadystate operation. The pulses of dead polymer will experience a residence time distribution in the CSTR given by ##EQU43## where the pulse of dead polymer is formed at time t=0.
Let us consider the consumption of dead polymer chains with ilong chain branches per polymer molecule born at time t=t and also the distortion of the weight chain length distribution of these chains, giving W(r, i, t, t). ##EQU44## P is the total polymeric radical constant. Y_{0} is the total polymeric radical concentration. For a CSTR at steadystate, Applicants assume that k_{trp} and Y_{0} are constants and integrate Equation C analytically to give: ##EQU45##
Equation R gives the concentration of dead polymer of chain length r and with i trifunctional long chain branches for the pulse of dead polymer formed at time t at some later time t in the CSTR operating at steadystate. In the time interval t to t, chains of P(r,i,t,t) are removed from the family residing in the CSTR by chain transfer to polymer and outflow from the CSTR. Chain transfer to polymer distorts P(r,i,t,t) while outflow reduces P(r,i,t,t) but does not distort W(r,i,t,t), the weight chain length distribution of this pulse of dead polymer. P(r,i,t,t) is the concentration of the pulse of dead polymer formed at time t which remains in the CSTR at some later time t. The total concentration of dead polymer with chain length r and having i trifunctional long chain branches per polymer molecule in the CSTR at steadystate is given by ##EQU46## where R(r,i,t,t) is the rate of formation of dead polymer with chain length r and having i trifunctional long chain branches per polymer molecule at steadystate in moles/unit of time and is independent of time t. θ is the mean residence time of the CSTR. Note that
P(r,i,t,t)=R(r,i,t,t)dt/V in moles/dm.sup.3
Given W(t,t) and assuming that dead polymer chains for each branched family, i=1,2,3, . . . when produced instantaneously follow the SchulzZimm distribution of chain length, one may readily calculate P(r,i,t,t) from the moments mentioned earlier.
Let us now consider two CSTR reactors of equal volume in series operating at steadystate. Now one must account for the fact that say dead polymer chains formed as a pulse in CSTR(1) at time t will spend part of the time interval t to t in CSTR(1) and some part of time in CSTR(2) with the remaining having flowed out of CSTR(2). It has already been shown how to calculate P(r,i,t,t) and P(r,i) in CSTR(1) the upstream CSTR reactor. It is more difficult to calculate P(r,i) for the downstream CSTR reactor. We will use subscripts on these quantities to differentiate where they are located, either in CSTR(1) or CSTR(2). For example:
P_{1} (r,i,t,t) is the concentration of the dead polymer pulse formed at time t, in CSTR(1), in moles/dm^{3} and P_{2} (r,i,t,t) is the same but for CSTR(2).
P_{11} (r,i,t,t) is the concentration of the dead polymer pulse formed at time t in CSTR(1), at some later time t in CSTR(1).
P_{12} (r,i,t,t) is the concentration of the dead polymer pulse formed at time t in CSTR(1), at some later time t in CSTR(2).
P_{22} (r,i,t,t) is the concentration of the dead polymer pulse formed at time t in CSTR(2), at some later time t in CSTR(2).
It is difficult to calculate P_{12} (r,i,t,t) because one must account for these chains, flowing into CSTR(2), being consumed by chain transfer to polymer in CSTR(2) and flowing out of CSTR(2). The following population balance equation applies. ##EQU47## where
v_{1} and v_{2} are outlet total volumetric flowrates for CSTR(1) and CSTR(2), respectively
K_{trp} and Y_{0} are the chain transfer to polymer rate constant and total polymeric radical concentration values, respectively, for CSTR(2).
We have an analytical solution for P_{11} (r,i,t,t) and therefore one can obtain an analytical solution for P_{12} (r,i,t,t). An analytical solution is also available for P_{22} (r,i,t,t). One can therefore readily calculate P_{2} (r,i) the concentration of dead polymer with chain length r and having i trifunctional long chain branches in CSTR(2) at steady state.
Thus, ##EQU48## Therefore ##EQU49## where ##EQU50## which is solved using integration factor Φ as follows.
ΦdP.sub.12 (r,i,t,t)+ΦA*P.sub.12 (r,i,t,t)dt=ΦB exp (C*t)dt
dΦP.sub.12 (r,i,t,t)=ΦdP.sub.12 (r,i,t,t)+P.sub.12 (r,i,t,t)dΦ
therefore dΦ=ΦAdt and Φ=exp (At).
Further ##EQU51## when t=t,
P.sub.12 (r,i,t,t)=0
thus ##EQU52## and ##EQU53## which is the concentration of dead polymer from a pulse formed at time t in CSTR(1) at some later time t flowing out of CSTR(2) and into CSTR(3) . The solution of P_{12} (r,i,t,t) is then used to calculate the concentration of dead polymer from the pulse formed at time t in CSTR(1) at some later time t in CSTR(3), and so on for subsequent CSTR's.
Thus, it is the foregoing calculations of P(r,i,t,t) and P(r,i) in a first and subsequent reactors, that provide/enable the present invention tracking of polymer properties and property distributions throughout (in time and location) a polymer manufacturing system.
While this invention has been particularly shown and described with references to preferred embodiments thereof, it will be understood by those skilled in the art that various changes in form and details may be made therein without departing from the spirit and scope of the invention as defined by the appended claims. Those skilled in the art will recognize or be able to ascertain using no more than routine experimentation, many equivalents to the specific embodiments of the invention described specifically herein. Such equivalents are intended to be encompassed in the scope of the claims.
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